Stochastic programs where the uncertainty distribution must be inferred from noisy data samples are considered. The stochastic programs are approximated with distributionally-robust optimizations that minimize the worst-case expected cost over ambiguity sets, i.e., sets of distributions that are sufficiently compatible with the observed data. In this paper, the ambiguity sets capture the set of probability distributions whose convolution with the noise distribution remains within a ball centered at the empirical noisy distribution of data samples parameterized by the total variation distance. Using the prescribed ambiguity set, the solutions of the distributionally-robust optimizations converge to the solutions of the original stochastic programs when the numbers of the data samples grow to infinity. Therefore, the proposed distributionally-robust optimization problems are asymptotically consistent. This is proved under the assumption that the distribution of the noise is uniformly diagonally dominant. More importantly, the distributionally-robust optimization problems can be cast as tractable convex optimization problems and are therefore amenable to large-scale stochastic problems.

翻译：考虑噪声数据样本中推断不确定性分布的随机规划问题。通过分布鲁棒优化近似求解此类问题，该优化方法最小化模糊集上的最坏情况期望成本，其中模糊集指与观测数据充分相容的概率分布集合。本文提出的模糊集捕获了以下概率分布族：这些分布与噪声分布的卷积结果，始终位于以数据样本经验噪声分布为中心、由全变差距离参数化的球内。采用该模糊集时，随着数据样本数量趋于无穷，分布鲁棒优化的解收敛于原始随机规划的解，因此所提出的分布鲁棒优化问题具有渐近一致性。在噪声分布满足一致对角占优假设的条件下，该收敛性得到证明。更重要的是，此类分布鲁棒优化问题可转化为易处理的凸优化问题，因而适用于大规模随机问题的求解。